Letter, from Brook Taylor to [John Keill] Opening paragraph establishes a new mode of address between Taylor and Keill, leaving compliments aside. Discusses methods of approximation: Keill has convinced Taylor that his method of rational calculation is good but still he opts for the irrational form because [Edmund] Halley used it in his exraction of roots at the end of Is[aac] N[ewton]'s algebra. Builds on [Jacob] Bernouilli's series or quadratures. Thanks Keill for his "very ingenious demonstration of the velocity of a fluid running out of a vessel" although it was not what he was asking for "to prove that the velocity ought to be generated in one instant""For I can't understand how velocity can be generated in any particle of matter in an instant, unless it be by the impulse of a body moving with certain velocity, or by an elastical spring that unbends itself in the shortest time imaginable"proposes a potential experiment to investigate this. [Clarification: Recipient noted in handwritten note likely by William Young FRS, and can be deduced from address "Mr Professor" and content of letter] 9 October 1712